On the crossing relation in the presence of defects

TL;DR

This paper investigates the operator product expansion in defect-laden conformal field theories, explicitly computing duality transformations and connecting correlator channels to Wilson and 't Hooft operators within Liouville theory and the AGT framework.

Contribution

It provides explicit calculations of duality transformations of 4-point functions with defects in rational and Liouville CFTs, linking defect correlators to gauge theory operators.

Findings

Explicit duality transformation formulas for defect correlators.

Reproduction of Wilson and 't Hooft operator expectation values.

Connection established between defect CFT correlators and AGT conjecture insights.

Abstract

The OPE of local operators in the presence of defect lines is considered both in the rational CFT and the $c>25$ Virasoro (Liouville) theory. The duality transformation of the 4-point function with inserted defect operators is explicitly computed. The two channels of the correlator reproduce the expectation values of the Wilson and 't Hooft operators, recently discussed in Liouville theory in relation to the AGT conjecture.